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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

The space of isometry covariant tensor valuations


Authors: D. Hug, R. Schneider and R. Schuster
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 137-158
MSC (2000): Primary 52A20
Published electronically: December 17, 2007
MathSciNet review: 2319515
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Abstract | References | Similar Articles | Additional Information

Abstract: It is known that the basic tensor valuations which, by a result of S. Alesker, span the vector space of tensor-valued, continuous, isometry covariant valuations on convex bodies, are not linearly independent. P. McMullen has discovered linear dependences between these basic valuations and has implicitly raised the question as to whether these are essentially the only ones. The present paper provides a positive answer to this question. The dimension of the vector space of continuous, isometry covariant tensor valuations, of a fixed rank and of a given degree of homogeneity, is explicitly determined. The approach is constructive and permits one to provide a specific basis.


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Additional Information

D. Hug
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany
Email: daniel.hug@math.uni-freiburg.de

R. Schneider
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany
Email: rolf.schneider@math.uni-freiburg.de

R. Schuster
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg i. Br., Germany
Email: raschuster@munichre.com

DOI: http://dx.doi.org/10.1090/S1061-0022-07-00990-9
PII: S 1061-0022(07)00990-9
Keywords: Convex body, tensor valuation, isometry covariance
Received by editor(s): August 1, 2006
Published electronically: December 17, 2007
Additional Notes: Supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953
Dedicated: Dedicated to Professor Viktor Abramovich Zalgaller on the occasion of his 85th birthday
Article copyright: © Copyright 2007 American Mathematical Society